On the Canonical Foliation of an Indefinite Locally Conformal Kähler Manifold with a Parallel Lee Form

We study the semi-Riemannian geometry of the foliation F of an indefinite locally conformal Kähler (l.c.K.) manifold M, given by the Pfaffian equation ω=0, provided that ∇ω=0 and c=∥ω∥≠0 (ω is the Lee form of M). If M is conformally flat then every leaf of F is shown to be a totally geodesic semi-Riemannian hypersurface in M, and a semi-Riemannian space form of sectional curvature c/4, carrying an indefinite c-Sasakian structure. As a corollary of the result together with a semi-Riemannian version of the de Rham decomposition theorem any geodesically complete, conformally flat, indefinite Vaisman manifold of index 2s, 0<s<n, is locally biholomorphically homothetic to an indefinite complex Hopf manifold CHsn(λ), 0<λ<1, equipped with the indefinite Boothby metric gs,n.

Small Gauge Transformations and Universal Geometry in Heterotic Theories

The first part of this paper describes in detail the action of small gauge transformations in heterotic supergravity. We show a convenient gauge fixing is 'holomorphic gauge' together with a condition on the holomorphic top form. This gauge fixing, combined with supersymmetry and the Bianchi identity, allows us to determine a set of non-linear PDEs for the terms in the Hodge decomposition. Although solving these in general is highly non-trivial, we give a prescription for their solution perturbatively in α and apply this to the moduli space metric. The second part of this paper relates small gauge transformations to a choice of connection on the moduli space. We show holomorphic gauge is related to a choice of holomorphic structure and Lee form on a 'universal bundle'. Connections on the moduli space have field strengths that appear in the second order deformation theory and we point out it is generically the case that higher order deformations do not commute.

Infinitesimal Transformations of Locally Conformal Kähler Manifolds

The article is devoted to infinitesimal transformations. We have obtained that LCK-manifolds do not admit nontrivial infinitesimal projective transformations. Then we study infinitesimal conformal transformations of LCK-manifolds. We have found the expression for the Lie derivative of a Lee form. We have also obtained the system of partial differential equations for the transformations, and explored its integrability conditions. Hence we have got the necessary and sufficient conditions in order that the an LCK-manifold admits a group of conformal motions. We have also calculated the number of parameters which the group depends on. We have proved that a group of conformal motions admitted by an LCK-manifold is isomorphic to a homothetic group admitted by corresponding Kählerian metric. We also established that an isometric group of an LCK-manifold is isomorphic to some subgroup of the homothetic group of the coresponding local Kählerian metric.

Infinitesimal Transformations of Locally Conformal Kähler Manifolds

The article is devoted to infinitesimal transformations. We have obtained that LCK-manifolds do not admit nontrivial infinitesimal projective transformations. Then we study infinitesimal conformal transformations of LCK-manifolds. We have found the expression for the Lie derivative of a Lee form. Also we have obtained the system of partial differential equations for the transformations, and explored its integrability conditions. Hence we have got the necessary and sufficient conditions in order that the an LCK-manifold admits a group of conformal motions. Also we have calculated the number of parameters which the group depends on. We have proved that a group of conformal motions admitted by an LCK-manifold is isomorphic to a homothetic group admitted by corresponding K\"{a}hlerian metric. We also established that an isometric group of an LCK-manifold is isomorphic to a some subgroup of homothetic group of the coresponding local K\"{a}hlerian metric.

Locally conformally Kähler solvmanifolds: a survey

A Hermitian structure on a manifold is called locally conformally Kähler (LCK) if it locally admits a conformal change which is Kähler. In this survey we review recent results of invariant LCK structures on solvmanifolds and present original results regarding the canonical bundle of solvmanifolds equipped with a Vaisman structure, that is, a LCK structure whose associated Lee form is parallel.

Example of a six-dimensional LCK solvmanifold

The purpose of this paper is to prove that there exists a lattice on a certain solvable Lie group and construct a six-dimensional locally conformal Kähler solvmanifold with non-parallel Lee form.

CR-submanifolds with the symmetric σ in a locally conformal Kaehler space form

In this paper, we consider CR-submanifolds with the symmetric ?? which is a generalization of parallel second fundamental form, in a locally conformal Kaehler space form. About the symmetric tensor field P defined in (1.7), we show that, in an anti-holomorphic submanifold in an l.c.K.-space form, P is diagonal with respect to an adapted frame and has two eigenfunctions (See Theorem 3.1). Finally, we consider the relation of the eigenfunctions of P and the Lee form (See Theorems 3.2 and 3.3).

On the Application of Fourier Decomposition Parameters

The application of Fourier decomposition parameters has revolutionized important areas of investigations of Cepheid type variables since the introduction of Fourier analysis in its modern form by Simon and Lee (1981).In the literature several different representations of the results of Fourier analysis have been utilized. In view of the growing interest for applications of Fourier decomposition it is important to use and publish Fourier data in an optimal way. Most studies until now have used amplitude ratios and phase differences derived from traditional light curves giving the light variation in magnitudes, following the original recipe of Simon and Lee (1981). However, Stellingwerf and Donohoe (1986) advocated the use of phases rather than phase differences. Recently, Buchler et al. (1990) argued that the standard Simon & Lee form contains all relevant physics, and suggested analysis of flux-values rather than of magnitudes, because this removes the distorting effects of constant, false light. Thus there are many choices to be made in practical applications of Fourier analysis, and there is at present no convincing argument for preferring one specific representation.